Introduction

In this poster, I am seeking to answer the question “does income affect happiness?” using data from the General Social Survey (GSS). Since 1972, the GSS has conducted a yearly survey on adults in the United States that asks questions related to attitudes, behaviors, and opinions. The data set also includes several demographic variables about each respondent (Davern et al. 2022).

Data and methodology

I filtered the GSS data set down to the following 3 relevant variables:

• AGE: Age of Respondent
• HAPPY: Respondent’s General Happiness Level (3 Factors: Very Happy, Pretty Happy, or Not Too Happy)
• REALRINC: Respondent’s Real Income in Constant 1986 Dollars

Below are some summary statistics for the chosen variables:

I have transformed the REALRINC by applying a logarithmic transformation. This normalizes the variable and makes visualizations much more intuitive, allowing us to analyze percent changes in income rather than absolute changes (Kahneman and Deaton 2010). Below is a visual of the transformation:

All observations in GSS are assumed to be independent. The HAPPY variable is coded such that 1 corresponds to Very Happy, 2 corresponds to Pretty Happy, and 3 corresponds to Not Too Happy. Below is a simple distribution of the HAPPY data.

This analysis uses a mix of numerical (REALRINC and AGE) and categorical (HAPPY) variables. Since HAPPY has three levels, I decided that using a density plot was best to show how HAPPY changes as REALRINC and AGE increase. This is included in the next section.

I have decided to use a logistic regression model to predict values of HAPPY. From the plot in the next section, it is clear to see that the HAPPY score of 2, corresponding to Pretty Happy, is relatively uniform across all levels of log(REALRINC). As a result, I have grouped together levels 2 and 3, so the outcome of the regression model is a binary variable. We will call this new outcome happiness_boolean, where a value of 0 corresponds to Not Too Happy or Pretty Happy and a value of 1 corresponds to Very Happy. Below is the equation I will be using:

\[ E(Y_i | log(REALRINC), AGE) = \\ \frac{1}{1 + e^{(-(\beta_0 + \beta_1(log(REALRINC)) + \beta_2(AGE))}} \]

where \(E(Y_i | log(REALRINC), AGE)\) is the expected value of happiness_boolean based on log(REALRINC) and AGE. In this model, each \(\beta\) is a coefficient in the regression model. I will interpret each coefficient in context in the next section.

Results

I first decided to plot log(REALRINC) and HAPPY using a density graph. This shows a clear trend between an increasing income and happiness levels.

I have also chosen to show summary statistics of log income at each happiness level using the boxplots below.

From this density plot, we can see the pattern between AGE and HAPPY.

It is clear to see the relationship between HAPPY and REALRINC, but less clear to see a relationship between HAPPY and AGE. Let’s run a regression and see why this is.

Returning to our regression equation, we find that

\[ E(Y_i | log(REALRINC), AGE) = \\ \frac{1}{1 + e^{(-(-2.33328 + 0.12819(log(REALRINC)) + 0.0073046(AGE))}} \]

Here is a plot of the logistic model, as well as the values of happiness_boolean.

Discussion

• When both log(REALRINC) and AGE equal zero (which is impossible, as we saw), the probability of being Very Happy, rather than Pretty Happy or Not So Happy, is \(\frac{e^{\beta_0}}{1 + e^{\beta_0}} = \frac{e^{-2.3332862}}{1 + e^{-2.3332862}} = 0.088403\)
• There is a clear and statistically significant (t value = 11.722635) connection between HAPPY and log(REALRINC). From the regression analysis, we found that \(\beta_1 = 0.12819\), so \(exp(0.12819) \approx 1.136769\). This means that when a person’s log(REALRINC) increases by one unit, the the odds that person is Very Happy, rather than Pretty Happy or Not So Happy, increases by 13.6769%.
• There is a statistically significant relationship between HAPPY and AGE (t value = 8.395572), but the relationship is a small one. We found that \(\beta_2 = 0.0073046\) and so \(exp(0.0073046) = 1.007331\). From this we conclude that when a person’s AGE increases by one unit, the odds that the person is Very Happy, rather than Pretty Happy or Not So Happy, increases by 0.7331%.
• It is important to note the goodness-of-fit of this model. Even though the coefficients are statically significant, the AUC of the ROC curve is about 0.6, meaning that this is not a great fit for the model.
• Both of these results correspond with the literature that I found, showing that there is likely a positive relationship between happiness and both age and real income.